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  <div class="section" id="numpy-random-generator-standard-t">
<h1>numpy.random.Generator.standard_t<a class="headerlink" href="#numpy-random-generator-standard-t" title="Permalink to this headline">¶</a></h1>
<p>method</p>
<dl class="method">
<dt id="numpy.random.Generator.standard_t">
<code class="sig-prename descclassname">Generator.</code><code class="sig-name descname">standard_t</code><span class="sig-paren">(</span><em class="sig-param">df</em>, <em class="sig-param">size=None</em><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.Generator.standard_t" title="Permalink to this definition">¶</a></dt>
<dd><p>Draw samples from a standard Student’s t distribution with <em class="xref py py-obj">df</em> degrees
of freedom.</p>
<p>A special case of the hyperbolic distribution.  As <em class="xref py py-obj">df</em> gets
large, the result resembles that of the standard normal
distribution (<a class="reference internal" href="numpy.random.standard_normal.html#numpy.random.standard_normal" title="numpy.random.standard_normal"><code class="xref py py-obj docutils literal notranslate"><span class="pre">standard_normal</span></code></a>).</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>df</strong><span class="classifier">float or array_like of floats</span></dt><dd><p>Degrees of freedom, must be &gt; 0.</p>
</dd>
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Output shape.  If the given shape is, e.g., <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">n,</span> <span class="pre">k)</span></code>, then
<code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">k</span></code> samples are drawn.  If size is <code class="docutils literal notranslate"><span class="pre">None</span></code> (default),
a single value is returned if <code class="docutils literal notranslate"><span class="pre">df</span></code> is a scalar.  Otherwise,
<code class="docutils literal notranslate"><span class="pre">np.array(df).size</span></code> samples are drawn.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray or scalar</span></dt><dd><p>Drawn samples from the parameterized standard Student’s t distribution.</p>
</dd>
</dl>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>The probability density function for the t distribution is</p>
<div class="math">
<p><img src="../../../_images/math/0d804954793333ce66b9c157bac58c6c368e6106.svg" alt="P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df}
\Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}"/></p>
</div><p>The t test is based on an assumption that the data come from a
Normal distribution. The t test provides a way to test whether
the sample mean (that is the mean calculated from the data) is
a good estimate of the true mean.</p>
<p>The derivation of the t-distribution was first published in
1908 by William Gosset while working for the Guinness Brewery
in Dublin. Due to proprietary issues, he had to publish under
a pseudonym, and so he used the name Student.</p>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="rb7c952f3992e-1"><span class="brackets"><a class="fn-backref" href="#id3">1</a></span></dt>
<dd><p>Dalgaard, Peter, “Introductory Statistics With R”,
Springer, 2002.</p>
</dd>
<dt class="label" id="rb7c952f3992e-2"><span class="brackets">2</span></dt>
<dd><p>Wikipedia, “Student’s t-distribution”
<a class="reference external" href="https://en.wikipedia.org/wiki/Student's_t-distribution">https://en.wikipedia.org/wiki/Student’s_t-distribution</a></p>
</dd>
</dl>
<p class="rubric">Examples</p>
<p>From Dalgaard page 83 <a class="reference internal" href="#rb7c952f3992e-1" id="id3">[1]</a>, suppose the daily energy intake for 11
women in kilojoules (kJ) is:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">intake</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">5260.</span><span class="p">,</span> <span class="mi">5470</span><span class="p">,</span> <span class="mi">5640</span><span class="p">,</span> <span class="mi">6180</span><span class="p">,</span> <span class="mi">6390</span><span class="p">,</span> <span class="mi">6515</span><span class="p">,</span> <span class="mi">6805</span><span class="p">,</span> <span class="mi">7515</span><span class="p">,</span> \
<span class="gp">... </span>                   <span class="mi">7515</span><span class="p">,</span> <span class="mi">8230</span><span class="p">,</span> <span class="mi">8770</span><span class="p">])</span>
</pre></div>
</div>
<p>Does their energy intake deviate systematically from the recommended
value of 7725 kJ?</p>
<p>We have 10 degrees of freedom, so is the sample mean within 95% of the
recommended value?</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">()</span><span class="o">.</span><span class="n">standard_t</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">100000</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">intake</span><span class="p">)</span>
<span class="go">6753.636363636364</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">intake</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">ddof</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="go">1142.1232221373727</span>
</pre></div>
</div>
<p>Calculate the t statistic, setting the ddof parameter to the unbiased
value so the divisor in the standard deviation will be degrees of
freedom, N-1.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">intake</span><span class="p">)</span><span class="o">-</span><span class="mi">7725</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">intake</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">ddof</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">intake</span><span class="p">)))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</pre></div>
</div>
<p>For a one-sided t-test, how far out in the distribution does the t
statistic appear?</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">s</span><span class="o">&lt;</span><span class="n">t</span><span class="p">)</span> <span class="o">/</span> <span class="nb">float</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
<span class="go">0.0090699999999999999  #random</span>
</pre></div>
</div>
<p>So the p-value is about 0.009, which says the null hypothesis has a
probability of about 99% of being true.</p>
<div class="figure align-default">
<img alt="../../../_images/numpy-random-Generator-standard_t-1.png" src="../../../_images/numpy-random-Generator-standard_t-1.png" />
</div>
</dd></dl>

</div>


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